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Spreading and shortest paths in systems with sparse long-range connections. (English) Zbl 1062.82507

Summary: Spreading according to simple rules (e.g., of fire or diseases) and shortest-path distances are studied on d-dimensional systems with a small density p per site of long-range connections (‘small-world’ lattices). The volume V(t) covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t. From this, the average shortest-path distance l(r) can be calculated as a function of Euclidean distance r. It is found that l(r)r for

r<r c =[2pΓ d (d-1)!] -1/d log(2pΓ d L d )

and l(r)r c for r>r c . The characteristic length r c , which governs the behavior of shortest-path lengths, diverges logarithmically with L for all p>0.

MSC:
82C20Dynamic lattice systems and systems on graphs
92D30Epidemiology